GRE Physics Companion: Coupled Oscillators and Normal Modes

GRE Physics Companion: Coupled Oscillators and Normal Modes

For a conservative linear system,

(1)

Normal modes satisfy

(2)

and

(3)

1 High-value GRE facts

1. A normal mode has one common angular frequency and fixed amplitude ratios.
2. Normal frequencies follow from det(K − ω²M) = 0.
3. Mode shapes are generalized eigenvectors.
4. For two equal masses with outer springs k and coupling spring k_c, ω₁² = k∕m and ω₂² = (k + 2k _c)∕m.
5. The in-phase mode does not stretch the coupling spring.
6. The out-of-phase mode has the larger frequency for positive coupling.
7. General motion is a superposition of normal modes.
8. Distinct modes are orthogonal with respect to the mass matrix.
9. zero modes represent motions with no restoring force.
10. Weak coupling produces closely spaced frequencies and slow beating.

Part I: Original GRE-style problems

Problem 1: normal-mode condition

A normal mode of

(4)

satisfies

1. Ka = 0 only
2. (K − ω²M)a = 0
3. Ma = 0
4. q = 0 only
5. K + ωM = 0

Problem 2: equal-mass symmetric mode

For two equal masses with wall springs k and coupling spring k_c, the in-phase normal frequency is

1. 
2. 
3. 
4. 
5. zero

Problem 3: antisymmetric mode

For the same system, the out-of-phase normal frequency is

1. 
2. 
3. 
4. 
5. zero

Problem 4: symmetric mode shape

A mode shape for the lower-frequency equal-mass mode is proportional to

1. 
2. 
3. 
4. 
5. 

Problem 5: normal coordinates

For equal masses, a convenient symmetric normal coordinate is

1. (x₁ + x₂)∕
2. (x₁ − x₂)∕
3. x₁x₂
4. x₁∕x₂
5. x₁² + x ₂²

Problem 6: mode superposition

An arbitrary small motion of a conservative linear coupled system can generally be written as

1. one normal mode only
2. a sum of normal modes
3. a constant displacement only
4. one exponential only
5. a rigid translation only

Problem 7: zero mode

Two equal masses are connected only by one spring and otherwise free. The in-phase translation mode has

1. ω = 0
2. ω =
3. ω =
4. infinite frequency
5. imaginary frequency

Problem 8: mass orthogonality

For distinct normal modes of a symmetric conservative system,

1. a_r^T Ma _s = 0
2. a_r^T Ka _s = 1
3. a_r = a_s
4. M = 0
5. ω_r = ω_s

Problem 9: weak coupling

If

(5)

the mode splitting is approximately

1. ω₀k∕k_c
2. ω₀k_c∕k
3. k_c∕m
4. 
5. zero exactly

Problem 10: numerical frequencies

For

(6)

the lower normal frequency is

1. 2 rad/s
2. 3 rad/s
3. 4 rad/s
4. rad/s
5. 9 rad/s

Problem 11: same numerical system

For the system in Problem 10, the upper normal frequency is

1. 3 rad/s
2. 4 rad/s
3. rad/s
4. rad/s
5. 17 rad/s

Problem 12: continuum bridge

A long chain of coupled oscillators is important because

1. all normal frequencies become zero
2. it approaches the normal-mode description of a continuous wave system
3. coupling disappears
4. the mass matrix becomes irrelevant
5. every mode has the same frequency

Part II: Complete worked solutions

Solution 1

Using

(7)

gives

(8)

Answer: (B).

Solution 2

In the in-phase mode,

(9)

so the coupling spring is not stretched. Only the wall springs provide restoring force:

(10)

Answer: (A).

Solution 3

In the out-of-phase mode, the coupling spring contributes additional restoring stiffness:

(11)

Answer: (C).

Solution 4

The lower-frequency mode is the in-phase motion:

(12)

Answer: (C).

Solution 5

The symmetric normal coordinate is

(13)

Answer: (A).

Solution 6

The normal modes form a basis for the linear small-oscillation motion. Therefore a general motion is a superposition of modes. Answer: (B).

Solution 7

Equal translation does not stretch the only spring, so there is no restoring force:

(14)

Answer: (A).

Solution 8

For distinct eigenfrequencies of a symmetric conservative system,

(15)

Answer: (A).

Solution 9

For weak coupling,

(16)

Answer: (B).

Solution 10

(17)

Answer: (B).

Solution 11

ω2	=	(18)
	=  rad∕s.	(19)

Answer: (C).

Solution 12

As the number of coupled oscillators becomes large, the discrete mode family approaches the normal modes of a continuous medium and hence wave mechanics. Answer: (B).

2 GRE checklist

1. Write the coupled equations before guessing mode shapes.
2. Form M and K and solve the generalized eigenvalue problem.
3. Find eigenvectors as well as eigenvalues.
4. Use symmetry to recognize in-phase and out-of-phase modes quickly.
5. Transform both displacement and velocity initial data into normal coordinates.
6. Remember that distinct modes are mass orthogonal.
7. Check for zero modes when the system has an unconstrained symmetry motion.
8. For weak coupling, use the normal-frequency splitting to estimate beat time.

References

References